Spinor Groups with Good Reduction

TL;DR

This paper proves finiteness results for spinor groups with good reduction over certain 2-dimensional global fields, extending to related groups and establishing key cohomological finiteness properties.

Contribution

It establishes the finiteness of isomorphism classes of spinor groups with good reduction over 2-dimensional global fields, extending to unitary and G2 groups, and proves related cohomological finiteness.

Findings

Finiteness of isomorphism classes of spinor groups with good reduction.

Finiteness of the genus of such groups.

Properness of the global-to-local map in Galois cohomology.

Abstract

Let $K$ be a 2-dimensional global field of characteristic $\neq 2$, and let $V$ be a divisorial set of places of $K$. We show that for a given $n \geqslant 5$, the set of $K$-isomorphism classes of spinor groups $G = \mathrm{Spin}_n(q)$ of nondegenerate $n$-dimensional quadratic forms over $K$ that have good reduction at all $v \in V$, is finite. This result yields some other finiteness properties, such as the finiteness of the genus $\mathbf{gen}_K(G)$ and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups $H^i(K , \mu_2)_V$ for $i \geqslant 1$ established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type $\textsf{G}_2$.